I fundamentally agree with your last statement FD, we already do have moral questions examined in a more scientific fashion, in "Ethics", and it is studied along philosophical lines.
I have watched the video again, so I will have another go at putting across what Harris actually said, and will quote him in some places.
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My precis of what he said, take two.
FACTS & VALUES are not necessarily belonging to different spheres, we derive our values from observing facts.
He defines "Values" as "Facts" about the wellbeing of conscious creatures, and that all values are reducible to a concern about conscious experience.
The condition of the "consciousness of wellbeing", can be observed through a "continuum of facts".
Observation let's us observe truths we can agree on.
These truths revealed by observation, show us that there are right and wrong answers to how humans flourish.
He then goes on and describes the moral landscape as having many peaks and troughs, so there can be more than one "right" answer to questions.
He uses nutrition, and health as analogies, with nutrition he points out there are an abundance of good foods we can choose, not just one, so because we may be able to define many positive moral precepts, does not mean we cannot identify negative ones, which range from foods that are just not as good for us, through to things that are poisonous, and these negative values occupy the moral troughs, and do not aid humans in flourishing.
He then says how religious demagogues agree with him that we need a universal conception of human values, but they declare their values are from divine sources, and he thinks we could do better by using critical analysis of observation, by utilising a "domain of expertise".
The domain of expertise term, is saying that not everybody's opinion is as valid as the next guy's, he asks if we would consult the Taliban on questions of physics, and asks is their attitudes to advancing human wellbeing any more worthy of consideration in the field of morals, than their knowledge of physics is to science?
In all fields there are those who stand out as leading experts, and morals should be no different, in recognising, and valuing their opinions more highly.
He finishes his talk with graphics showing both extremist muslims and christians each holding up placards depicting violent, hateful messages, and says we have been told we need to respect and understand different cultures, and different societies, but we cannot allow that to then stop us from addressing whether or not what they are offering is right or wrong.
To demand tolerance for what we know is wrong makes no more sense than demanding we tolerate no disease control standards for society.
To address that we must first agree that we can define what is right or wrong, that we are capable of defining the answers to moral questions.